It is known that one-dimensional (1D) strongly correlated systems in condensed matter physics exhibit a large number of exotic behaviors, and their mysterious properties can be simulated in well-controlled quantum simulators constructed by optical systems of cold atoms and photons. One promising example of quantum optical simulators is the 1D optical nonlinear waveguide with the tight field confinement and coherent photon trapping techniques, where dark-state polaritons are formed as light-matter excitations. It is the purpose of our work to efficiently observe some well-known models and phenomena in condensed matter physics using polaritons in nonlinear waveguides. These polaritons are shown to follow the dynamics of a quantum system like the Lieb-Liniger model, the Bose-Hubbard model, the quantum sine-Gordon model, the Fermi-Hubbard model, and the relativistic Thirring model. We analyze possible experimental detections of various characteristic correlations of polaritons/photons, which can be used to resolve some long-standing problems.